The Condensate from Torus Knots

TL;DR

This paper explores the duality between instantons and torus knots in 5D supersymmetric QED, revealing how knot invariants relate to instanton condensates and their embedding into 4D theories.

Contribution

It introduces a novel connection between instanton ensembles, torus knot invariants, and geometric transitions in supersymmetric gauge theories.

Findings

HOMFLY invariants encode instanton and electric charge degeneracies.

Embedding into SU(2) SQCD links knot theory with gauge theory dualities.

Relation to Wilson loops in 3D Chern-Simons theory is established.

Abstract

We discuss recently formulated instanton-torus knot duality in $\Omega$-deformed 5D SQED on $\mathbb{R}^4 \times S^1$ focusing at the microscopic aspects of the condensate formation in the instanton ensemble. Using the chain of dualities and geometric transitions we embed the SQED with a surface defect into the $SU(2)$ SQCD with $N_f=4$ and identify the numbers $(n,m)$ of the torus $T_{n,m}$ knot as instanton charge and electric charge. The HOMFLY torus knot invariants in the fundamental representation provide entropic factor in the condensate of the massless flavor counting the degeneracy of the instanton--W-boson web with instanton and electric numbers $(n,m)$ but different spin and flavor content. Using the inverse geometrical transition we explain how our approach is related to the evaluation of the HOMFLY invariants in terms of Wilson loop in 3d CS theory. The reduction to 4D theory is briefly considered and some analogy with baryon vertex is conjectured.